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Question Number 2432 by prakash jain last updated on 19/Nov/15
What is the sum of digits of 3333^(4444) ,  Say sum of all digits of 3333^(4444)  is A,  If A>10 then sum all digits of A.  This process is repeated until a single  digits sum x in obtained.  x=?
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} , \\ $$$$\mathrm{Say}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} \:\mathrm{is}\:\mathrm{A}, \\ $$$$\mathrm{If}\:\mathrm{A}>\mathrm{10}\:\mathrm{then}\:\mathrm{sum}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{A}. \\ $$$$\mathrm{This}\:\mathrm{process}\:\mathrm{is}\:\mathrm{repeated}\:\mathrm{until}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{digits}\:\mathrm{sum}\:{x}\:\mathrm{in}\:\mathrm{obtained}. \\ $$$${x}=? \\ $$
Answered by prakash jain last updated on 20/Nov/15
3333≡3(mod 9)  3333^2 ≡9(mod 9)  3333^(4444) ≡9(mod 9)  x=9
$$\mathrm{3333}\equiv\mathrm{3}\left(\mathrm{mod}\:\mathrm{9}\right) \\ $$$$\mathrm{3333}^{\mathrm{2}} \equiv\mathrm{9}\left(\mathrm{mod}\:\mathrm{9}\right) \\ $$$$\mathrm{3333}^{\mathrm{4444}} \equiv\mathrm{9}\left(\mathrm{mod}\:\mathrm{9}\right) \\ $$$${x}=\mathrm{9} \\ $$

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